Elliptic Curve Cryptography

1. ELLIPTIC CURVE CRYPTOGRAPHY
~ S. Janani, AP/CSE, KCET
CS8792 – CRYPTOGRAPHY
AND NETWORK SECURITY

2. ECC Diffie-Hellman
 can do key exchange analogous to D-H
 users select a suitable curve Eq(a,b)
 select base point G=(x1,y1)
 with large order n s.t. nG=O
 A & B select private keys nA<n, nB<n
 compute public keys: PA=nAG, PB=nBG
 compute shared key: K=nAPB, K=nBPA
 same since K=nAnBG
 attacker would need to find k, hard

3. ECC Encryption/Decryption
 several alternatives, will consider simplest
 must first encode any message M as a point on the
elliptic curve Pm
 select suitable curve & point G as in D-H
 each user chooses private key nA<n
 and computes public key PA=nAG
 to encrypt Pm : Cm={kG, Pm+kPb}, k random
 decrypt Cm compute:
Pm+kPb–nB(kG) = Pm+k(nBG)–nB(kG) = Pm

4. ECC Security
 can use much smaller key sizes than with RSA
etc
 for equivalent key lengths computations are
roughly equivalent
 ECC offers significant computational
advantages

5. Comparable Key Sizes for
Equivalent Security
Symmetric scheme
(key size in bits)
ECC-based scheme
(size of n in bits)
RSA/DSA
(modulus size in
bits)
56 112 512
80 160 1024
112 224 2048
128 256 3072
192 384 7680
256 512 15360

6. Summary
 have considered:
 Elliptic Curves
 Elliptic Curve Arithmetic
 Elliptic Curve cryptography
• Key Exchange(Diffie Hellman)
• Encryption and Decryption